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54 ROBERT F. BROWN AND HELGA SCHIRMER
Definition 2.1. Let f : M → N be a map of manifolds. Then three types
of maps are defined as follows.
(1) Type I: f is orientation-true.
(2) Type II: f is not orientation-true but does not map an orientationreversing
loop in M to a contractible loop in N.
(3) Type III: f maps an orientation-reversing loop in M to a contractible
loop in N.
Further, a map f is defined to be orientable if it is of Type I or II, and
non-orientable otherwise.
The term orientable map is sometimes used for Type I, that is orientation-true,
maps; see [Do, Exercise 6, p. 271]. The characterisation of the
three types is based on Olum [O, p. 475] (see also [Sk, p. 416]). An
equivalent characterisation in terms of the orientability of covering spaces
of M and N, which we will use in §3, is given by Epstein [E, p. 371]. In
essence, the characterisation of the three types of maps is already contained
in Hopf’s paper [H2]. We shall see that maps of the first two types share
many properties with regard to Nielsen root theory and, as Hopf was well
aware of this, he considered maps of these first two types together and
therefore introduced the concept of an orientable or non-orientable map
[H2, Definition V, p. 579].
The following examples illustrate the three types of maps.
Example 2.2 (Type I). (a) If M and N are orientable manifolds, then all
maps f : M → N are orientation-true. (b) For N a non-orientable manifold
and M its orientable covering, the covering map p: M → N is of Type I.
(The case of N the projective plane is mentioned by Hopf [H2, p. 584].)
(c) The identity map of a non-orientable manifold is an example of a Type
I map between non-orientable manifolds.
Example 2.3 (Type II). Let M+ be the Möbius band and let p: M+ → S 1
be the fibration obtained by retracting M+ to its central circle. Let i: S 1 →
S 1 ×I = N+ be defined by setting i(x) = (x, 0), then f+ = i◦p is a boundarypreserving
map from the Möbius band to the annulus. Let f = 2f+ : M =
2M+ → 2N+ = N be the double of the map f+, so M is the Klein bottle and
N is the torus. The loops representing elements in the kernel of the induced
homomorphism fπ : π1(M, x0) → π1(N, c) are orientation-preserving, so f is
not Type III. Since a map from a non-orientable manifold to an orientable
manifold cannot be orientation-true, we conclude that f is Type II.
Example 2.4 (Type III). (a) For M a non-orientable manifold, a constant
map f : M → N is obviously of Type III. (b) For an example of a Type III
map of M onto N, let T 2 denote the torus and P 2 the projective plane. Let
D be a disc in T 2 and let id: T 2 −int D → T 2 −int D be the identity. Extend
id to f : T 2 #P 2 → T 2 by extending the identity map on ∂D in P 2 − int D